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      【SDOI2014】数表
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        <p>几天前写的题目忘记写题解了……倒回来思考了半天才想起来</p>
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<p>先假设没有不大于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span></span></span></span>这条恶心的限制，设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span></span></span></span>表示<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span></span></span></span>的约数和，约定<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>≤</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\leq m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span><span class="mrel">≤</span><span class="mord mathit">m</span></span></span></span>，则我们要求的就是：</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtable><mtr><mtd><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mi>f</mi><mo>(</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo><mo>)</mo></mrow></mtd><mtd><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mo>[</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo><mo>=</mo><mo>=</mo><mi>d</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow></mrow></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow></mrow></msubsup><mo>[</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo><mo>=</mo><mo>=</mo><mn>1</mn><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow></mrow></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow></mrow></msubsup><msub><mo>∑</mo><mrow><mi>t</mi><mi mathvariant="normal">∣</mi><mi>g</mi><mi>c</mi><mi>d</mi><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo></mrow></msub><mi>μ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo><msubsup><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mrow><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow></mrow></msubsup><mi>μ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow></mrow></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{aligned}
	\sum_{i=1}^n\sum_{j=1}^mf(gcd(i,j))&amp;=\sum_{d=1}^nf(d)\sum_{i=1}^n\sum_{j=1}^m[gcd(i,j)==d]\\
	&amp;=\sum_{d=1}^nf(d)\sum_{i=1}^{\left \lfloor \frac{n}{d} \right \rfloor}\sum_{j=1}^{\left \lfloor \frac{m}{d} \right \rfloor}[gcd(i,j)==1]\\
	&amp;=\sum_{d=1}^nf(d)\sum_{i=1}^{\left \lfloor \frac{n}{d} \right \rfloor}\sum_{j=1}^{\left \lfloor \frac{m}{d} \right \rfloor}\sum_{t \vert gcd(i,j)}\mu (t)\\
	&amp;=\sum_{d=1}^nf(d)\sum_{t=1}^{\left \lfloor \frac{n}{d} \right \rfloor}\mu (t) \left\lfloor \frac{n}{dt} \right\rfloor \left\lfloor \frac{m}{dt} \right\rfloor
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:7.2900420000000015em;"></span><span class="strut bottom" style="height:14.080084000000003em;vertical-align:-6.7900420000000015em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist"><span style="top:-5.638645em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.2500050000000003em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000254801em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.2500050000000005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">m</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mord mathit">c</span><span class="mord mathit">d</span><span class="mopen">(</span><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">)</span><span class="mclose">)</span></span></span><span style="top:-1.9638630000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"></span></span><span style="top:1.7109190000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"></span></span><span style="top:5.487929000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="col-align-l"><span class="vlist"><span style="top:-5.638645em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">d</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.2500050000000003em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000254801em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.2500050000000005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">m</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">[</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mord mathit">c</span><span class="mord mathit">d</span><span class="mopen">(</span><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">)</span><span class="mrel">=</span><span class="mrel">=</span><span class="mord mathit">d</span><span class="mclose">]</span></span></span><span style="top:-1.9638630000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">d</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.4635050000000003em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="minner scriptstyle uncramped"><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌊</span><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">d</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌋</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000254801em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.4635050000000005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="minner scriptstyle uncramped"><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌊</span><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">d</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit">m</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌋</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mopen">[</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mord mathit">c</span><span class="mord mathit">d</span><span class="mopen">(</span><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">)</span><span class="mrel">=</span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mclose">]</span></span></span><span style="top:1.7109190000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">d</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.4635050000000003em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="minner scriptstyle uncramped"><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌊</span><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">d</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌋</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000254801em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.4635050000000005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="minner scriptstyle uncramped"><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌊</span><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">d</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit">m</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌋</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mop op-limits"><span class="vlist"><span style="top:1.241005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">t</span><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mord mathit">c</span><span class="mord mathit">d</span><span class="mopen">(</span><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">μ</span><span class="mopen">(</span><span class="mord mathit">t</span><span class="mclose">)</span></span></span><span style="top:5.487929000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">d</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span><span class="mop op-limits"><span class="vlist"><span style="top:1.1671129999999998em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">t</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.4635050000000003em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="minner scriptstyle uncramped"><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌊</span><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">d</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-scriptstyle textstyle uncramped" style="top:0.07500000000000001em;">⌋</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mord mathit">μ</span><span class="mopen">(</span><span class="mord mathit">t</span><span class="mclose">)</span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌊</span></span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathit">d</span><span class="mord mathit">t</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌋</span></span></span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌊</span></span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathit">d</span><span class="mord mathit">t</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathit">m</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌋</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span></span></span></span></p>
<p>设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi><mo>=</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">T=dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="mrel">=</span><span class="mord mathit">d</span><span class="mord mathit">t</span></span></span></span>，得到：</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>T</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow><mrow><mo fence="true">⌊</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>T</mi></mrow></mfrac><mo fence="true">⌋</mo></mrow><msub><mo>∑</mo><mrow><mi>d</mi><mi mathvariant="normal">∣</mi><mi>T</mi></mrow></msub><mi>μ</mi><mrow><mo fence="true">(</mo><mfrac><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo fence="true">)</mo></mrow><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\sum_{T=1}^n \left\lfloor \frac{n}{T} \right\rfloor \left\lfloor \frac{m}{T} \right\rfloor \sum_{d \vert T}\mu \left (\frac{T}{d} \right ) f(d)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.6513970000000002em;"></span><span class="strut bottom" style="height:3.167402em;vertical-align:-1.516005em;"></span><span class="base displaystyle textstyle uncramped"><span class="mop op-limits"><span class="vlist"><span style="top:1.1943359999999998em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.2500050000000003em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌊</span></span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌋</span></span></span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌊</span></span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathit">m</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size2">⌋</span></span></span><span class="mop op-limits"><span class="vlist"><span style="top:1.241005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">d</span><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">μ</span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathit">d</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span></span></span></span></span></p>
<p>于是可以设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>d</mi><mi mathvariant="normal">∣</mi><mi>T</mi></mrow></msub><mi>μ</mi><mrow><mo fence="true">(</mo><mfrac><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo fence="true">)</mo></mrow><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">h(T)=\sum_{d \vert T}\mu  \left  (\frac{T}{d}  \right  )  f(d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.872331em;"></span><span class="strut bottom" style="height:1.347341em;vertical-align:-0.47501em;"></span><span class="base textstyle uncramped"><span class="mord mathit">h</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop"><span class="op-symbol small-op mop" style="top:-0.0000050000000000050004em;">∑</span><span class="vlist"><span style="top:0.30001em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">d</span><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">μ</span><span class="minner textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">d</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span></span></span></span>，那么求出<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">h</span></span></span></span>的前缀和就可以整除分块做了</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span></span></span></span>数组可以线性筛预处理出，当然你要是像我一样懒，写一个<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mi>n</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathit">n</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span>的暴力搞出<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span></span></span></span>数组也行</p>
<p>现在来考虑不大于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span></span></span></span>这条限制</p>
<p>我们可以把所有询问离线，然后按<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span></span></span></span>的大小排序，优先处理<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span></span></span></span>较小的询问。再把<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span></span></span></span>数组离线，按值升序排序，并保存下标。然后用一个树状数组来维护<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">h</span></span></span></span>的前缀和。每次处理一个询问，对于所有不大于当前<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span></span></span></span>的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span></span></span></span>，把<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>μ</mi><mo>(</mo><mi>T</mi><mi mathvariant="normal">/</mi><mi>d</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\mu(T/d)f(d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">μ</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="mord mathrm">/</span><span class="mord mathit">d</span><span class="mclose">)</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span></span></span></span>给加到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">h(T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">h</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span></span>里面去。这一步骤用筛法来做就是：枚举<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span>，对于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi><mi>d</mi><mo>&lt;</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">id&lt;N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span><span class="mord mathit">d</span><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>的情况，让<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi><mo>(</mo><mi>i</mi><mi>d</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">h(id)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">h</span><span class="mopen">(</span><span class="mord mathit">i</span><span class="mord mathit">d</span><span class="mclose">)</span></span></span></span>加上<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>μ</mi><mo>(</mo><mi>i</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\mu(i)f(d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">μ</span><span class="mopen">(</span><span class="mord mathit">i</span><span class="mclose">)</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">d</span><span class="mclose">)</span></span></span></span>。</p>
<p>然后整除分块时就可以直接在树状数组里求h的前缀和</p>
<p>时间复杂度<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mi>Q</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(Q\sqrt{n}log  n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8002800000000001em;"></span><span class="strut bottom" style="height:1.05028em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathit">Q</span><span class="sqrt mord"><span class="sqrt-sign" style="top:0.03971999999999998em;"><span class="style-wrap reset-textstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord mathit">n</span></span></span><span style="top:-0.72028em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit">o</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span></p>
<div class="highlight-box" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false" contenteditable="true" data-rel="CPP"><figure class="iseeu highlight /cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> INF 0x7fffffff</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> lowbit(x) (x&amp;-x)</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">100000</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">QRY</span>&#123;</span><span class="keyword">int</span> n,m,a,id;&#125; qry[<span class="number">20010</span>];</span><br><span class="line">pair&lt;<span class="keyword">int</span>,<span class="keyword">int</span>&gt; f[N+<span class="number">10</span>];</span><br><span class="line"><span class="keyword">int</span> prime[<span class="number">9592</span>+<span class="number">10</span>],tot=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">int</span> mu[N+<span class="number">10</span>];</span><br><span class="line"><span class="keyword">bool</span> mark[N+<span class="number">10</span>];</span><br><span class="line"><span class="keyword">int</span> bit[N],ans[N];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">Add</span><span class="params">(<span class="keyword">int</span> p,<span class="keyword">int</span> x)</span></span>&#123;<span class="keyword">for</span>(;p&lt;=N;p+=lowbit(p))bit[p]+=x;&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">Sum</span><span class="params">(<span class="keyword">int</span> p)</span></span>&#123;<span class="keyword">int</span> res=<span class="number">0</span>;<span class="keyword">for</span>(;p&gt;<span class="number">0</span>;p-=lowbit(p))res+=bit[p];<span class="keyword">return</span> res;&#125;</span><br><span class="line"><span class="keyword">bool</span> <span class="keyword">operator</span> &lt; (<span class="keyword">const</span> QRY &amp;a,<span class="keyword">const</span> QRY &amp;b)&#123;<span class="keyword">return</span> a.a&lt;b.a;&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">Init</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">	mu[<span class="number">1</span>]=<span class="number">1</span>;</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">2</span>;i&lt;=N;i++)</span><br><span class="line">	&#123;</span><br><span class="line">		<span class="keyword">if</span>(!mark[i]) prime[++tot]=i,mu[i]=<span class="number">-1</span>;</span><br><span class="line">		<span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">1</span>;j&lt;=tot&amp;&amp;i*prime[j]&lt;=N;j++)</span><br><span class="line">		&#123;</span><br><span class="line">			mark[i*prime[j]]=<span class="number">1</span>;</span><br><span class="line">			<span class="keyword">if</span>(i%prime[j]) mu[i*prime[j]]=-mu[i];</span><br><span class="line">			<span class="keyword">else</span> mu[i*prime[j]]=<span class="number">0</span>;</span><br><span class="line">		&#125;</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=N;i++)</span><br><span class="line">		<span class="keyword">for</span>(<span class="keyword">int</span> j=i;j&lt;=N;j+=i)</span><br><span class="line">			f[j].first+=i;</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=N;i++) f[i].second=i;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">	Init();</span><br><span class="line">	<span class="keyword">int</span> Q;</span><br><span class="line">	<span class="built_in">scanf</span>(<span class="string">"%d"</span>,&amp;Q);</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=Q;i++)</span><br><span class="line">	&#123;</span><br><span class="line">		<span class="built_in">scanf</span>(<span class="string">"%d%d%d"</span>,&amp;qry[i].n,&amp;qry[i].m,&amp;qry[i].a);</span><br><span class="line">		qry[i].id=i;</span><br><span class="line">	&#125;</span><br><span class="line">	sort(qry+<span class="number">1</span>,qry+<span class="number">1</span>+Q);</span><br><span class="line">	sort(f+<span class="number">1</span>,f+<span class="number">1</span>+N);</span><br><span class="line">	<span class="keyword">int</span> pos=<span class="number">1</span>;</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> T=<span class="number">1</span>;T&lt;=Q;T++)</span><br><span class="line">	&#123;</span><br><span class="line">		<span class="keyword">while</span>(pos&lt;=N&amp;&amp;f[pos].first&lt;=qry[T].a)</span><br><span class="line">		&#123;</span><br><span class="line">			<span class="keyword">int</span> k=f[pos].second;</span><br><span class="line">			<span class="keyword">for</span>(<span class="keyword">int</span> j=k;j&lt;=N;j+=k)</span><br><span class="line">				Add(j,f[pos].first*mu[j/k]);</span><br><span class="line">			pos++;</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">int</span> div,n=qry[T].n,m=qry[T].m,res=<span class="number">0</span>;</span><br><span class="line">		<span class="keyword">if</span>(n&gt;m) swap(n,m);</span><br><span class="line">		<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i=div+<span class="number">1</span>)</span><br><span class="line">		&#123;</span><br><span class="line">			div=min(n/(n/i),m/(m/i));</span><br><span class="line">			res+=(n/i)*(m/i)*(Sum(div)-Sum(i<span class="number">-1</span>));</span><br><span class="line">		&#125;</span><br><span class="line">		ans[qry[T].id]=res&amp;INF;</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=Q;i++) <span class="built_in">printf</span>(<span class="string">"%d\n"</span>,ans[i]);</span><br><span class="line">	<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></div>

      
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